|
|
|
||
|
We discuss the differential geometric concepts of symplectic and Poisson geometry, where smooth manifolds are
enhanced with symplectic forms and Poisson bivectors. The emerging notions of Hamiltonian vector field,
Marsden—Weinstein reduction and Poisson cohomology are considered. One of the main goals of the course is to
investigate the Fedosov construction of symplectic star products. The latter is a deformation quantization, which can
be applied to an arbitrary symplectic manifold.
Last update: Fatková Tereza, Ing. (02.01.2026)
|
|
||
|
After reviewing the basics of differential geometry, smooth manifolds, vector bundles, Cartan calculus, we discuss symplectic and Poisson manifolds. These are manifolds together with a (nondegenerate) Poisson bivector and this additional structure allows us to consider Poisson cohomology, Hamiltonian and Poisson vector fields. From a physical point of view, symplectic manifolds admit an interpretation in the context of Hamiltonian mechanics. We further describe a phase space reduction in terms of Marsden-Weinstein reduction and discuss a Lie algebroid interpretation of Poisson manifolds. In the second half of the course, we study deformation quantization, particularly of symplectic manifolds. The goal is to give a full proof of the celebrated Fedosov construction of star products for arbitrary symplectic manifolds. Last update: Weber Thomas, Ph.D. (31.01.2026)
|
|
||
|
The final exam consists of a seminar talk of about 45 minutes, presented by the participant. For this, a number of possible seminar topics will be given at the end of the course. The material will be based on the content of the course and is meant to extend the discussed material. For interested students there might be the possibility to continue the course project in form of a master thesis. Last update: Weber Thomas, Ph.D. (02.02.2025)
|
|
||
|
Symplectic geometry:
E. Meinrenken: Symplectic Geometry, lecture notes, University of Toronto, 2000. S. Waldmann: Poisson-Geometrie und Deformationsquantisierung, Springer, 2007.
Poisson geometry:
C. Esposito: Formality Theory: From Poisson structures to Deformation Quantization, Springer, 2015. A. Cannas da Silva, A. Weinstein: Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes. AMS, 1999.
Fedosov quantization:
J. Schnitzer: Poisson Geometry and Deformation Quantization, lecture notes, 2023. Last update: Weber Thomas, Ph.D. (31.01.2026)
|
|
||
|
Blackboard class with seminars at the end of the course.
The course will also be streamed via Teams. Recordings, as well as lecture notes, will be made available on Teams and via email. Last update: Weber Thomas, Ph.D. (02.02.2025)
|
|
||
|
1) Smooth manifolds
2) Vector bundles and the Cartan calculus
3) Symplectic geometry
4) Marsden-Weinstein reduction
5) Poisson geometry
6) Poisson cohomology and Lie algebroids
7) Deformation quantization
8) Fedosov quantization
This schedule is preliminary. Last update: Weber Thomas, Ph.D. (31.01.2026)
|